A fast algorithm for finding "small" solutions of F(x,y)=G(x,y) over imaginary quadratic fields

Abstract

Let d be a square-free positive integer, and K = $\text{Q}$($\text{i}\text{d}$) an imaginary quadratic field. Let F(X,Y), G(X,Y) ϵ $\text{Z}$_K[X,Y] be polynomials of degree n and m , respectively. Assume, that F is homogeneous, irreducible and n - m ≥ 4. We give a fast algorithm (based on the LLL basis reduction algorithm) for finding the solutions of the inequality

|F(X,Y)| ≤ |G(X,Y)| in X, Y ϵ $\text{Z}$_k |X|,|y| < C

where C is a prescribed constant. We illustrate the method by solving

|X⁸ + (1 + i )X²Y⁶ + (2 - i )XY⁷ + (4 + i)Y⁶| ≤ 50· |X² + Y²|

in X,Y ϵ Z[i] with max{|X|,|Y|} < 10²⁰⁰.